The rest of this paper is organized as follows. In section , we give a brief introduction to the half quadratic models, the Ambrosio-Tortorelli model, and the basic KL analysis. In section , we give an illustration of the motivation of our preconditioned framework and prove that all \(L(u,y)\)  of discussed models are KL functions. In section , we first prove the iterative sequence \(\lbrace (u^k,y^k)\rbrace \)  is bounded for each model. We then get the global convergence with the KL properties by [1]}, [2]}, [3]}, [4]}.
In section , we first give the detailed five-point stencils of the symmetric Gauss-Seidel iterations along with the preconditioners and the numerical experiments then follows. In section , we give some discussions and conclusions.
